Osaka Journal of Mathematics

The magnetic flow on the manifold of oriented geodesics of a three dimensional space form

Yamile Godoy and Marcos Salvai

Full-text: Open access

Abstract

Let $M$ be the three dimensional complete simply connected manifold of constant sectional curvature $0,1$ or $-1$. Let $\mathcal{L}$ be the manifold of all (unparametrized) complete oriented geodesics of $M$, endowed with its canonical pseudo-Riemannian metric of signature $(2,2)$ and Kähler structure $J$. A smooth curve in $\mathcal{L}$ determines a ruled surface in $M$. We characterize the ruled surfaces of $M$ associated with the magnetic geodesics of $\mathcal{L}$, that is, those curves $\sigma$ in $\mathcal{L}$ satisfying $\nabla_{\dot{\sigma}}\dot{\sigma}=J\dot{\sigma}$. More precisely: a time-like (space-like) magnetic geodesic determines the ruled surface in $M$ given by the binormal vector field along a helix with positive (negative) torsion. Null magnetic geodesics describe cones, cylinders or, in the hyperbolic case, also cones with vertices at infinity. This provides a relationship between the geometries of $\mathcal{L}$ and $M$.

Article information

Source
Osaka J. Math., Volume 50, Number 3 (2013), 749-763.

Dates
First available in Project Euclid: 27 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1380287431

Mathematical Reviews number (MathSciNet)
MR3129001

Zentralblatt MATH identifier
1286.53049

Subjects
Primary: 53C22: Geodesics [See also 58E10] 53C35: Symmetric spaces [See also 32M15, 57T15] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 53C50: Lorentz manifolds, manifolds with indefinite metrics

Citation

Godoy, Yamile; Salvai, Marcos. The magnetic flow on the manifold of oriented geodesics of a three dimensional space form. Osaka J. Math. 50 (2013), no. 3, 749--763. https://projecteuclid.org/euclid.ojm/1380287431


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